Collins an unused sub as Ireland fall just short against Denmark
The point means the Mick McCarthy's side finished third in Group D and will have to hope for success through the play-offs, which take place next March.
Conor Hourihane shot straight at Kasper Schmeichel in the first half, before Alan Browne volleyed wide.
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Hide AdIn the second half, Kiaran Clark headed over twice, before the Danes went ahead on 78 minutes, Martin Braithwaite stabbing home Henrik Dalsgaard's cross.
Matt Doherty levelled with five minutes to go, but Ireland couldn't grab a winner and could now be drawn against either Bosnia-Herzegovina, Wales or Slovakia.
A positive McCarthy said: "We can beat anyone in the play-offs.
"The players will go back to their clubs and we don't have a game until March.
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Hide Ad"I'll be getting round them and reminding them it has been a good campaign and that we can play like that, so why not play like that in March?
"I am immensely proud of the players.
"They pushed them right to the limit. If my players and my team leave everything on the pitch I will take the result as it comes so I will have to take the defeat.
"You just need that bit of luck. We had opportunities but I am not going to criticise the lads - they have been brilliant.
"We lost one game out of eight. I'm very proud of them."